Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: Heteroclinic orbits

The classical Melnikov method for heteroclinic orbits is extended theoretically to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation. We assume that the unperturbed system is a piecewise Hamiltonian system with a pair of heteroclinic orbits. The heteroclinic orbit transversally jumps across the first switching manifold by an impulsive effect and crosses the second switching manifold continuously. In effect, the trajectory of the corresponding perturbed system crosses the second switching manifold by applying the reset map describing the impact rule instantaneously. The random Melnikov process of such systems is then derived by measuring the distance of perturbed stable and unstable manifolds, and the criteria for the onset of chaos with or without noise excitation is established. In this derivation process, we overcome the difficulty that the derivation method of the corresponding homoclinic case cannot be directly used due to the difference between the symmetry of the homoclinic orbit and the asymmetry of the heteroclinic orbit. Finally, we investigate the complicated dynamics of a particular piecewise-smooth system with and without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations by this new extended method and numerical simulations. The numerical results verify the correctness of the theoretical results and demonstrate that this extended method is simple and effective for studying the dynamics of such systems.